Informal inference should be a foundation for formal inference
Wild, Pfannkuch, Regan, and Horton (2011) provide a rich illustration of how statistics has previously been taught in schools:
Often descriptive statistics has been the only diet for students up to the penultimate year of high school, to be followed by an attempt to force-feed statistical inference, with its mathematical underpinnings, concepts and reasoning in the final year. (p. 249)
In their review of the literature, Wild et al. (2010, as cited in Wild et al., 2011) note that students seem to build their inferential reasoning skills in disjoint and “incoherent” ways (p. 249). Students and teachers also struggle with determining whether their conclusions are being made about the samples they analyze or the populations they sample from. They add that research greatly supports the claim that building a strong understanding of informal inference translates directly to students’ abilities to understand formal inference. Indeed, Wild et al. (2011) point out:
Research strongly suggests (e.g., Chance et al., 2004) that large numbers of students fail to comprehend formal statistical inference when they do meet it at either school or introductory university level, and that they will continue to do so unless a much better job is done of laying essential conceptual foundations over a period of years before any attempt to teach formal inference is made. Otherwise there are simply too many ideas to be comprehended and interlinked all at once. (p. 249)
A significant number of authors agree with Wild et al. regarding the importance of laying conceptual foundations, particularly with informal inference, prior to introducing students to formal inference (Marzano, 2010; Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008; Zieffler, Garfield, delMas, & Reading, 2008; Garfield & Ben-Zvi, 2008; Watson, 2001; Watson & Moritz, 1999, National Council of Teachers of Mathematics, 2000). Marzano (2010) further notes that inference is a cognitive process that is foundational to the higher-order thinking skills we expect 21st century students to utilize. Authors agree that even young children can begin developing inference skills through an open-ended, data-driven approach, since making inferences, such as predicting what happens next in a story, comes naturally for them (Marzano, 2010; Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008). Zieffler et al. (2008) point out that because students often face difficulties understanding statistical inference, and because statistical inference incorporates many topics in statistics, “introducing informal inference early and revisiting the topic throughout a single course or curriculum across grades could provide students with multiple opportunities to build the conceptual framework needed to support inferential reasoning” (p. 46).
Tasks should promote learning and understanding of informal inference
Attention to task design and careful sequencing of topics is promoted to foster students’ development of inference skills students (Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008; Zieffler et al., 2008; Garfield & Ben-Zvi, 2008; Garfield & Ben-Zvi, 2007; Stohl & Tarr, 2002). Despite the lack of a common approach, Garfield and Ben-Zvi (2007) reference various sequences suggested by other authors, and later, Garfield and Ben-Zvi (2008) provide a tentative concept sequence for developing informal inference, yet they acknowledge that this may or may not be an ideal approach. Though Zieffler et al. (2008) agree that lesson and sequence design seem to have an impact on laying the foundation of informal inference, they point out that there are “many unanswered questions” in this area of research (p. 54).
While engaging in inference activities, attention must be paid to the context of the data being analyzed, as conclusions are made based on that context (Rossman & Chance, 1999; Garfield & Ben-Zvi, 2008; Paparistodemou & Meletiou-Mavrotheris, 2008; Tarr, Lee, & Rider, 2006; Rossman, 2008). By selecting tasks that emphasize the context of the problem, teachers help to build their students’ skills in “decision making in a variety of real-world situations” (Paparistodemou & Meletiou-Mavrotheris, 2008, p. 83). It is “in a variety of authentic purposeful contexts where [students] need these [different statistical] tools to make sense of the situation” (Paparistodemou & Meletiou-Mavrotheris, 2008, p. 102).
Students should also be exposed to diverse data collection methods to identify which methods support specific objectives and which methods make statistical inferences valid (Rossman & Chance, 1999; Tarr et al., 2006). Students should engage in simulation activities that allow them to model certain phenomena, so that they have experience collecting data they will analyze (Stohl & Tarr, 2002; Wild et al., 2011; Rossman & Chance, 1999). With this experience, students will have a better understanding of the phenomenon, be able to collect and analyze large samples using technology, and make more informed inferential arguments (Garfield & Ben-Zvi, 2008; Stohl & Tarr, 2002; Tarr et al, 2006; Wild et al., 2011; Rossman, 2008).
Sample size is another important consideration for students when they are learning to draw conclusions about a certain phenomenon. Students should engage in activities that allow them to observe differences in results based on small versus large sample sizes and how their reasoning might be affected based on those sample sizes (Tarr et al., 2006; Wild et al., 2011; Ben-Zvi, 2006). For instance, Tarr et al. (2006) state, “a critical idea in understanding statistics is the notion that larger samples yield more power in making inferences” (p. 142). Tarr et al. (2006) and Wild et al. (2011) expand on this idea by emphasizing that students must understand how samples are or are not representative of a population before they can begin building an understanding of statistical inference.
Closely related to the idea of sample size is how students engage in the process of sampling. Many authors feel that repeated sampling is necessary to make stronger inferential arguments (Rossman, 2008; Garfield & Ben-Zvi, 2008; Wild et al., 2011). Garfield and Ben-Zvi (2008) suggest “resampling methods are viewed as having the potential to offer a way to build informal inferences without focusing on the details of mathematics and formulas” (p. 270). Thus, careful attention must be paid to methods of sampling and the importance of obtaining multiple samples to draw appropriate conclusions. Students should have experiences drawing repeated samples to help them develop understanding of sampling variability and skills in reasoning based on that variability (Rossman, 2008; Garfield & Ben-Zvi, 2008; Wild et al., 2011).
Statistical technology tools allow students to view multiple representations of data to better draw conclusions and form their reasoning (Rossman & Chance, 1999; Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008; Stohl & Tarr, 2002; Tarr et al., 2006; Wild et al. 2011). Indeed, Wild et al. (2011) note that “technology provides exciting possibilities for changing the landscape of statistics education in schools in ways that could make it unrecognizable” (p. 248). This is because technology allows students to choose appropriate sample sizes, quickly generate large samples, see the data represented in a variety of ways, and update results after each trial (Tarr et al., 2006). Students can use software tools to explore data in an open-ended way, interact with the data, and use visual displays to help support their inferences (Stohl & Tarr, 2002). Programs such as TinkerPlots (Konold & Miller, 2005) allow students to easily and dynamically manipulate data, efficiently draw repeated samples, and receive immediate feedback to answer different contextual questions as they explore the data (Paparistodemou & Meletiou-Mavrotheris, 2008). The power of these technological tools is that students can begin to focus on the “underlying concepts and properties of inference procedures” and be freed from “computational drudgery”(Rossman & Chance, 1999, p. 299). Paparistodemou and Meletiou-Mavrotheris (2008) take this idea even further by noting that technology “provide[s] young learners with tools they can use to construct their own conceptual understanding of statistical concepts” (p. 102). Wild et al. (2011) summarize the impact technology can have and emphasizes the need for attention to it in statistical tasks:
For statistics education, technology is the ultimate game changer. Its biggest pedagogical implications come from the fact that it allows us to conceptualize, in ways that were previously unavailable, potentially providing access to concepts at much earlier stages of development. With creative approaches, school level statistics can become much more ambitious, exciting and useful. Determining what a changed landscape could look like will, however, require the creative engagement of both academia and the profession. (p. 248)
References
Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.
Garfield, J,. & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International Statistical Review. 75(3), 372-396.
Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching practice. New York: Springer.
Konold, C., & Miller, C. D. (2005). TinkerPlots: Dynamic Data Explorations [software, Version 1.0]. Emeryville, CA: Key Curriculum Press. Available from http://www.keycurriculum.com/products/tinkerplots
Marzano, R. J. (2010). Teaching inference. Educational Leadership, 67(7), 80-81.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Paparistodemou, E., & Meletiou-Mavrotheris, M. (2008). Developing young students’ informal inference skills in data analysis. Statistics Education Research Journal, 7(2). 83-106.
Rossman, A. J. (2008). Reasoning about informal statistical inference: One statistician’s view. Statistics Education Research Journal, 7(2). 5-19.
Rossman, A. J., & Chance, B. L. (1999). Teaching the reasoning of statistical inference: A "top ten" list. The College Mathematics Journal, 30(4), 297-305.
Stohl, H. & Tarr, J. E. (2002). Developing notions of inference using probability simulation tools. Journal of Mathematical Behavior, 21, 319-337.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C. Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National Council of Teachers of Mathematics.
Watson, J. M. (2001). Longitudinal development of inferential reasoning by school students. Educational Studies in Mathematics, 47(3). 337-372.
Watson, J. M., & Moritz, J. B. (1999). The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37(2), 145-168.
Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. J. (2011). Towards more accessible conceptions of statistical inference. Journal of the Royal Statistical Society. Series A, 174(2), 247-295.
Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on informal inferential reasoning. Statistics Education Research Journal, 7(2), 40-58.
Back to Inference Home
Informal inference should be a foundation for formal inference
Wild, Pfannkuch, Regan, and Horton (2011) provide a rich illustration of how statistics has previously been taught in schools:
Often descriptive statistics has been the only diet for students up to the penultimate year of high school, to be followed by an attempt to force-feed statistical inference, with its mathematical underpinnings, concepts and reasoning in the final year. (p. 249)
In their review of the literature, Wild et al. (2010, as cited in Wild et al., 2011) note that students seem to build their inferential reasoning skills in disjoint and “incoherent” ways (p. 249). Students and teachers also struggle with determining whether their conclusions are being made about the samples they analyze or the populations they sample from. They add that research greatly supports the claim that building a strong understanding of informal inference translates directly to students’ abilities to understand formal inference. Indeed, Wild et al. (2011) point out:
Research strongly suggests (e.g., Chance et al., 2004) that large numbers of students fail to comprehend formal statistical inference when they do meet it at either school or introductory university level, and that they will continue to do so unless a much better job is done of laying essential conceptual foundations over a period of years before any attempt to teach formal inference is made. Otherwise there are simply too many ideas to be comprehended and interlinked all at once. (p. 249)
A significant number of authors agree with Wild et al. regarding the importance of laying conceptual foundations, particularly with informal inference, prior to introducing students to formal inference (Marzano, 2010; Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008; Zieffler, Garfield, delMas, & Reading, 2008; Garfield & Ben-Zvi, 2008; Watson, 2001; Watson & Moritz, 1999, National Council of Teachers of Mathematics, 2000). Marzano (2010) further notes that inference is a cognitive process that is foundational to the higher-order thinking skills we expect 21st century students to utilize. Authors agree that even young children can begin developing inference skills through an open-ended, data-driven approach, since making inferences, such as predicting what happens next in a story, comes naturally for them (Marzano, 2010; Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008). Zieffler et al. (2008) point out that because students often face difficulties understanding statistical inference, and because statistical inference incorporates many topics in statistics, “introducing informal inference early and revisiting the topic throughout a single course or curriculum across grades could provide students with multiple opportunities to build the conceptual framework needed to support inferential reasoning” (p. 46).
Tasks should promote learning and understanding of informal inference
Attention to task design and careful sequencing of topics is promoted to foster students’ development of inference skills students (Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008; Zieffler et al., 2008; Garfield & Ben-Zvi, 2008; Garfield & Ben-Zvi, 2007; Stohl & Tarr, 2002). Despite the lack of a common approach, Garfield and Ben-Zvi (2007) reference various sequences suggested by other authors, and later, Garfield and Ben-Zvi (2008) provide a tentative concept sequence for developing informal inference, yet they acknowledge that this may or may not be an ideal approach. Though Zieffler et al. (2008) agree that lesson and sequence design seem to have an impact on laying the foundation of informal inference, they point out that there are “many unanswered questions” in this area of research (p. 54).
While engaging in inference activities, attention must be paid to the context of the data being analyzed, as conclusions are made based on that context (Rossman & Chance, 1999; Garfield & Ben-Zvi, 2008; Paparistodemou & Meletiou-Mavrotheris, 2008; Tarr, Lee, & Rider, 2006; Rossman, 2008). By selecting tasks that emphasize the context of the problem, teachers help to build their students’ skills in “decision making in a variety of real-world situations” (Paparistodemou & Meletiou-Mavrotheris, 2008, p. 83). It is “in a variety of authentic purposeful contexts where [students] need these [different statistical] tools to make sense of the situation” (Paparistodemou & Meletiou-Mavrotheris, 2008, p. 102).
Students should also be exposed to diverse data collection methods to identify which methods support specific objectives and which methods make statistical inferences valid (Rossman & Chance, 1999; Tarr et al., 2006). Students should engage in simulation activities that allow them to model certain phenomena, so that they have experience collecting data they will analyze (Stohl & Tarr, 2002; Wild et al., 2011; Rossman & Chance, 1999). With this experience, students will have a better understanding of the phenomenon, be able to collect and analyze large samples using technology, and make more informed inferential arguments (Garfield & Ben-Zvi, 2008; Stohl & Tarr, 2002; Tarr et al, 2006; Wild et al., 2011; Rossman, 2008).
Sample size is another important consideration for students when they are learning to draw conclusions about a certain phenomenon. Students should engage in activities that allow them to observe differences in results based on small versus large sample sizes and how their reasoning might be affected based on those sample sizes (Tarr et al., 2006; Wild et al., 2011; Ben-Zvi, 2006). For instance, Tarr et al. (2006) state, “a critical idea in understanding statistics is the notion that larger samples yield more power in making inferences” (p. 142). Tarr et al. (2006) and Wild et al. (2011) expand on this idea by emphasizing that students must understand how samples are or are not representative of a population before they can begin building an understanding of statistical inference.
Closely related to the idea of sample size is how students engage in the process of sampling. Many authors feel that repeated sampling is necessary to make stronger inferential arguments (Rossman, 2008; Garfield & Ben-Zvi, 2008; Wild et al., 2011). Garfield and Ben-Zvi (2008) suggest “resampling methods are viewed as having the potential to offer a way to build informal inferences without focusing on the details of mathematics and formulas” (p. 270). Thus, careful attention must be paid to methods of sampling and the importance of obtaining multiple samples to draw appropriate conclusions. Students should have experiences drawing repeated samples to help them develop understanding of sampling variability and skills in reasoning based on that variability (Rossman, 2008; Garfield & Ben-Zvi, 2008; Wild et al., 2011).
Statistical technology tools allow students to view multiple representations of data to better draw conclusions and form their reasoning (Rossman & Chance, 1999; Ben-Zvi, 2006; Paparistodemou & Meletiou-Mavrotheris, 2008; Stohl & Tarr, 2002; Tarr et al., 2006; Wild et al. 2011). Indeed, Wild et al. (2011) note that “technology provides exciting possibilities for changing the landscape of statistics education in schools in ways that could make it unrecognizable” (p. 248). This is because technology allows students to choose appropriate sample sizes, quickly generate large samples, see the data represented in a variety of ways, and update results after each trial (Tarr et al., 2006). Students can use software tools to explore data in an open-ended way, interact with the data, and use visual displays to help support their inferences (Stohl & Tarr, 2002). Programs such as TinkerPlots (Konold & Miller, 2005) allow students to easily and dynamically manipulate data, efficiently draw repeated samples, and receive immediate feedback to answer different contextual questions as they explore the data (Paparistodemou & Meletiou-Mavrotheris, 2008). The power of these technological tools is that students can begin to focus on the “underlying concepts and properties of inference procedures” and be freed from “computational drudgery”(Rossman & Chance, 1999, p. 299). Paparistodemou and Meletiou-Mavrotheris (2008) take this idea even further by noting that technology “provide[s] young learners with tools they can use to construct their own conceptual understanding of statistical concepts” (p. 102). Wild et al. (2011) summarize the impact technology can have and emphasizes the need for attention to it in statistical tasks:
For statistics education, technology is the ultimate game changer. Its biggest pedagogical implications come from the fact that it allows us to conceptualize, in ways that were previously unavailable, potentially providing access to concepts at much earlier stages of development. With creative approaches, school level statistics can become much more ambitious, exciting and useful. Determining what a changed landscape could look like will, however, require the creative engagement of both academia and the profession. (p. 248)
References
Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.
Garfield, J,. & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International Statistical Review. 75(3), 372-396.
Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching practice. New York: Springer.
Konold, C., & Miller, C. D. (2005). TinkerPlots: Dynamic Data Explorations [software, Version 1.0]. Emeryville, CA: Key Curriculum Press. Available from http://www.keycurriculum.com/products/tinkerplots
Marzano, R. J. (2010). Teaching inference. Educational Leadership, 67(7), 80-81.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Paparistodemou, E., & Meletiou-Mavrotheris, M. (2008). Developing young students’ informal inference skills in data analysis. Statistics Education Research Journal, 7(2). 83-106.
Rossman, A. J. (2008). Reasoning about informal statistical inference: One statistician’s view. Statistics Education Research Journal, 7(2). 5-19.
Rossman, A. J., & Chance, B. L. (1999). Teaching the reasoning of statistical inference: A "top ten" list. The College Mathematics Journal, 30(4), 297-305.
Stohl, H. & Tarr, J. E. (2002). Developing notions of inference using probability simulation tools. Journal of Mathematical Behavior, 21, 319-337.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C. Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National Council of Teachers of Mathematics.
Watson, J. M. (2001). Longitudinal development of inferential reasoning by school students. Educational Studies in Mathematics, 47(3). 337-372.
Watson, J. M., & Moritz, J. B. (1999). The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37(2), 145-168.
Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. J. (2011). Towards more accessible conceptions of statistical inference. Journal of the Royal Statistical Society. Series A, 174(2), 247-295.
Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on informal inferential reasoning. Statistics Education Research Journal, 7(2), 40-58.